Schooling behavior driven complexities in a fear-induced prey–predator system with harvesting under deterministic and stochastic environments

The well-being of humans is closely linked to the well-being of species in any ecosystem, but the relationship between humans and nature has changed over time as societies have become more industrialized. In order to ensure the future of our ecosystems, we need to protect our planet’s biodiversity. In this work, a prey–predator model with fear dropping prey’s birth as well as death rates and nonlinear harvesting, is investigated. In addition, we consider that the consumption rate of predators, i.e., the functional response, is dependent on schooling behavior of both species. We have investigated the local stability of the equilibrium points and different types of bifurcations, such as transcritical, saddle-node, Hopf and Bogdanov–Takens (BT). We find that consumption rate of predator, fear and harvesting effort give complex dynamics in the neighbourhood of BT-points. Harvesting effort has both stabilizing and destabilizing effects. There is bistability between coexistence and predator-free equilibrium points in the system. Further, we have studied the deterministic model in fluctuating environment. Simulation results of stochastic system includes time series solutions of one simulation run and corresponding phase portraits. Notably, several simulation runs are conducted to obtain time series solutions, histograms, and stationary distributions. Our findings exhibit that during stochastic processes, model species fluctuate around some average values of the deterministic steady-state for lower environmental disturbances. However, higher values of environmental disturbances lead the species to extinction.